## Solving the Dual

It is possible but difficult to solve this dual problem using a three-dimensional graph or the simplex method. However, because the primal problem has been solved already, information from this solution can be used to easily solve the dual. Remember that the solutions to the primal and dual of a single linear programming problem are complementary, and the following must hold:

Primal Objective Variable; X Dual Slack Variable; = 0 Primal Slack Variable^ X Dual Objective Variable^ = 0

In this linear programming problem,

Because both R and TV have nonzero solutions in the primal, the dual slack variables LR and Ljy must equal zero at the optimal solution. Furthermore, because there is excess audience exposure to the single marital status category in the primal solution, SS ^ 0, the related dual shadow price variable VS must also equal zero in the optimal solution. This leaves only VA and Vj as two unknowns in the two-equation system of dual constraints:

10,000VA + 10,000Vj = $ 6,000 20,000VA + 10,000Vj = $10,000

Subtracting the second constraint equation from the first gives

Substituting the value $0.40 for VA in either constraint equation produces a value of $0.20 for Vj. Finally, substituting the appropriate values for VA, Vj, and VS into the dual objective function gives a value of C* = $56,000 [= ($0.40 X 100,000) + ($0.20 X 80,000) + ($0 X 40,000)]. This is the same figure as the $56,000 minimum cost solution to the primal.

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